Subgroups — #30675088

Order 16768.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523513	C₂	1	1	yes	maximal, characteristic
2	#22553440	C₂	1	2	yes	maximal, complemented
3	#22553385	C₂	1	2	yes	maximal, complemented
4	#22553380	C₂	1	2	yes	maximal, complemented
5	(C₂²⋊D₁₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
6	#22553368	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553364	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553334	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#432×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523513	C₄	1	1	yes	characteristic
13	C₂×#14523496	C₂²	1	2	yes	—
14	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×D₁₃₁	C₂²	1	2	yes	—
16	C₄×Dic₂₆₂	C₄	1	1	yes	characteristic
17	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
18	C₄×D₅₂₄	C₄	1	1	yes	characteristic
19	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₃₁	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₃₁	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic
24	#16862205	C₄	1	1	yes	characteristic
25	#16862203	C₄	1	1	yes	characteristic
26	#16862203	C₄	1	1	yes	characteristic
27	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
28	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
29	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
30	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
31	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
32	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
33	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
36	#16862156	C₂²	1	2	yes	—
37	C₈×Dic₁₃₁	C₂²	1	2	yes	—
38	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
42	#97×C₁₃₁	C₂²	1	2	yes	—
43	#14523513	—	2	4	no	—
44	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
45	C₂×D₅₂₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
54	#14523496	—	4	2	no	—
55	C₂×C₁₃₁⋊C₈	—	4	2	no	—
56	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
57	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
58	C₈×D₁₃₁	—	4	2	no	—
59	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
61	Dic₁₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
62	Dic₁₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
63	Dic₁₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
64	Dic₁₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
70	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
81	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
82	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
85	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
86	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
88	Dic₂₆₂	—	2	2	no	—
89	D₅₂₄	—	2	2	no	—
90	C₂²⋊D₁₃₁	—	2	2	no	—
91	C₂²⋊D₁₃₁	M₁₆	1	4	yes	complemented
92	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₁₃₁	—	2	2	no	—
99	C₄×D₁₃₁	—	2	2	no	—
100	C₄×D₁₃₁	—	2	2	no	—
101	C₄×D₁₃₁	—	2	1	no	—
102	C₄×D₁₃₁	—	2	1	no	—
103	C₄×D₁₃₁	—	2	1	no	—
104	C₄×D₁₃₁	—	2	1	no	—
105	C₄×D₁₃₁	—	2	2	no	—
106	C₁₃₁⋊C₈	—	2	2	no	—
107	C₁₃₁⋊C₈	—	2	2	no	—
108	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	1	no	—
112	C₂×C₄×C₁₃₁	—	2	1	no	—
113	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
119	C₈×C₁₃₁	—	2	2	no	—
120	C₈×C₁₃₁	—	2	2	no	—
121	C₂²×C₁₃₁	—	2	1	no	—
122	C₂²×C₁₃₁	—	2	1	no	—
123	C₂²×C₁₃₁	C₂×M₁₆	1	2	yes	—
124	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₁₃₁	—	2	1	no	—
126	C₂×D₁₃₁	—	2	1	no	—
127	C₂×D₁₃₁	—	2	1	no	—
128	C₂×D₁₃₁	—	2	1	no	—
129	C₂×D₁₃₁	—	2	1	no	—
130	C₂×D₁₃₁	C₂×M₁₆	1	2	yes	—
131	C₄×C₁₃₁	—	2	1	no	—
132	C₄×C₁₃₁	—	2	1	no	—
133	C₄×C₁₃₁	—	2	1	no	—
134	C₄×C₁₃₁	—	2	1	no	—
135	C₄×C₁₃₁	#131	1	2	yes	—
136	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
138	Dic₁₃₁	—	2	1	no	—
139	Dic₁₃₁	—	2	1	no	—
140	Dic₁₃₁	—	2	1	no	—
141	Dic₁₃₁	—	2	1	no	—
142	Dic₁₃₁	—	2	1	no	—
143	Dic₁₃₁	C₂×M₁₆	1	2	yes	—
144	C₂×C₁₃₁	—	2	2	no	—
145	C₂×C₁₃₁	#520	1	1	yes	characteristic
146	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
147	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
148	D₁₃₁	—	2	2	no	—
149	D₁₃₁	—	4	1	no	—
150	C₁₃₁	#2990	1	1	yes	characteristic, complemented, Sylow
151	#2990	—	131	1	no	maximal, Sylow
152	C₄×#41	—	131	1	no	—
153	#446	—	131	2	no	—
154	#435	—	131	2	no	—
155	#435	—	131	2	no	—
156	C₈×D₄	—	131	2	no	—
157	#432	—	131	1	no	—
158	#432	D₁₃₁	1	1	yes	characteristic, complemented
159	#431	—	131	1	no	—
160	#423	—	131	1	no	—
161	C₂×C₄⋊C₈	—	131	1	no	—
162	C₂×C₄⋊C₈	—	131	1	no	—
163	C₂×#41	—	131	1	no	—
164	C₂×M₁₆	—	131	2	no	—
165	C₂²×C₈	—	131	2	no	—
166	C₂²×C₈	—	131	2	no	—
167	C₄×Q₈	—	131	1	no	—
168	C₄×Q₈	—	131	1	no	—
169	C₄×D₄	—	131	1	no	—
170	C₄×D₄	—	131	1	no	—
171	C₄×D₄	—	131	1	no	—
172	C₄×D₄	—	131	1	no	—
173	C₄×D₄	—	131	1	no	—
174	C₄×D₄	—	131	1	no	—
175	#117	—	131	1	no	—
176	#117	—	131	1	no	—
177	#117	—	131	1	no	—
178	C₂×C₄²	—	131	1	no	—
179	C₂×C₄²	—	131	1	no	—
180	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
181	C₄⋊C₈	—	131	2	no	—
182	C₄⋊C₈	—	131	2	no	—
183	C₄⋊C₈	—	131	1	no	—
184	C₄⋊C₈	—	131	1	no	—
185	C₄⋊C₈	—	131	2	no	—
186	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
187	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
188	C₂²⋊C₈	—	131	2	no	—
189	C₂²⋊C₈	—	131	2	no	—
190	C₂²⋊C₈	C₂×D₁₃₁	1	2	yes	—
191	#97	—	131	2	no	—
192	#97	C₂×D₁₃₁	1	2	yes	—
193	C₄×C₈	—	131	2	no	—
194	#41	—	262	4	no	—
195	C₂×Q₈	—	131	1	no	—
196	C₂×D₄	—	131	1	no	—
197	C₂×D₄	—	131	1	no	—
198	C₂×D₄	—	131	1	no	—
199	C₂²×C₄	—	131	1	no	—
200	C₂²×C₄	—	131	1	no	—
201	C₂²×C₄	—	131	1	no	—
202	C₂²×C₄	—	131	1	no	—
203	C₂²×C₄	—	131	1	no	—
204	C₂²×C₄	—	131	1	no	—
205	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
208	M₁₆	—	524	2	no	—
209	C₂×C₈	—	524	2	no	—
210	C₂×C₈	—	524	2	no	—
211	C₂×C₈	—	131	2	no	—
212	C₂×C₈	—	131	2	no	—
213	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
214	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
215	C₄⋊C₄	—	131	1	no	—
216	C₄⋊C₄	—	131	1	no	—
217	C₄⋊C₄	—	131	1	no	—
218	C₄⋊C₄	—	131	1	no	—
219	C₄⋊C₄	—	131	1	no	—
220	C₄⋊C₄	—	131	1	no	—
221	C₂²⋊C₄	—	131	1	no	—
222	C₂²⋊C₄	—	131	1	no	—
223	C₂²⋊C₄	—	131	1	no	—
224	C₂²⋊C₄	—	131	1	no	—
225	C₂²⋊C₄	—	131	1	no	—
226	C₂²⋊C₄	—	131	1	no	—
227	C₄²	—	131	1	no	—
228	C₄²	—	131	1	no	—
229	C₄²	—	131	1	no	—
230	C₄²	—	131	1	no	—
231	C₄²	—	131	1	no	—
232	C₄²	—	131	1	no	—
233	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
234	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
235	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
236	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
237	C₂³	—	131	1	no	—
238	C₂³	—	131	1	no	—
239	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
240	Q₈	—	262	2	no	—
241	D₄	—	262	2	no	—
242	D₄	—	262	2	no	—
243	D₄	—	131	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	262	2	no	—
249	C₂×C₄	—	262	2	no	—
250	C₂×C₄	—	262	2	no	—
251	C₂×C₄	—	262	2	no	—
252	C₂×C₄	—	262	1	no	—
253	C₂×C₄	—	262	1	no	—
254	C₂×C₄	—	262	1	no	—
255	C₂×C₄	—	262	1	no	—
256	C₂×C₄	—	131	1	no	—
257	C₂×C₄	—	131	1	no	—
258	C₂×C₄	—	131	1	no	—
259	C₂×C₄	—	131	1	no	—
260	C₂×C₄	—	131	1	no	—
261	C₂×C₄	—	131	1	no	—
262	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	262	2	no	—
271	C₈	—	262	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	262	1	no	—
275	C₂²	—	262	1	no	—
276	C₂²	—	262	1	no	—
277	C₂²	—	262	1	no	—
278	C₂²	—	262	1	no	—
279	C₂²	—	131	2	no	—
280	C₂²	M₁₆×D₁₃₁	1	2	yes	—
281	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	262	1	no	—
287	C₄	—	262	1	no	—
288	C₄	—	262	1	no	—
289	C₄	—	262	1	no	—
290	C₄	—	262	1	no	—
291	C₄	—	131	2	no	—
292	C₄	#16862254	1	2	yes	—
293	C₄	#16862292	1	1	yes	characteristic
294	C₄	#16862293	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	262	2	no	—
297	C₂	—	524	1	no	—
298	C₂	#22553792	1	1	yes	characteristic
299	C₂	C₂×M₁₆×D₁₃₁	1	1	yes	characteristic
300	C₂	C₂×#16862254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#432×C₁₃₁	C₂
prime radical	C₂×#14523513	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#432	D₁₃₁
hypocenter	C₁₃₁	#2990
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22553440	C₂	semidirect
3. #22553385	C₂	semidirect
4. #22553380	C₂	semidirect
5. (C₂²⋊D₁₃₁)⋊C₈	C₂	semidirect
6. #22553368	C₂	semidirect
7. #22553364	C₂	semidirect
8. #22553334	C₂	semidirect
9. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
10. C₄⋊C₈×D₁₃₁	C₂	semidirect
11. #432×C₁₃₁	C₂	semidirect
91. C₂²⋊D₁₃₁	M₁₆	semidirect
150. C₁₃₁	#2990	semidirect
158. #432	D₁₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14523513	2
#22553440	2
#22553385	2
#22553380	2
(C₂²⋊D₁₃₁)⋊C₈	2
#22553368	2
#22553364	2
#22553334	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
C₄⋊C₈×D₁₃₁	2
#432×C₁₃₁	2
#2990	131